# Understanding Permutation and Combination -1

If I go back to my previous post of finding traveling salesman problem, and I need to know how many options are available to visit all the cities, understanding of permutations and combinations can help me.

Permutation:
In how many ways we can choose and arrange options. For example, you have to create N digit number as a code, say 4 digit number.

You can choose the number in 10^4 ways. This is how-

First digit- 10 options (0 to 9 – say 0 is allowed)
Second digit- 10 options
and so on.

So at the end we have
10*10*10*10 options or 10^4

or to generalize

n*n*n*n.. r times i.e. n^r

The above case is for permutation where repetition is allowed. There can be a case where repetition is not allowed, e.g. instead of choosing numbers randomly, we are picking random balls from a sack. So if you have picked ball number 7 once, it will not be available for reuse.

First digit- 10 options
Second digit- 9 options (first digit option not available for reuse).
Third- 8
Fourth- 7

Total Options 10*9*8*7

generalize

n*(n-1)*(n-2).. (n-r)

nPr = n!/(n-r)!